A converse to a theorem of Adamyan, Arov and Krein
J.~Agler and N.~J.~Young
Keywords
interpolation, meromorphic function, pseudomultiplier
Status
J. Amer. Math. Soc. 12(1999) 305--333.
Abstract
A well known theorem of Akhiezer, Adamyan, Arov and Krein gives a
criterion (in terms of the signature of a certain Hermitian matrix)
for interpolation by a meromorphic function in the unit disc with
at most $m$ poles subject to an $L^\infty$-norm bound on the
unit circle.
One can view this theorem as an assertion about the Hardy space $H^2$
of analytic functions on the disc and its reproducing kernel.
A similar assertion makes sense
(though it is not usually true) for an
{\em arbitrary} Hilbert space of functions. One can therefore
ask for which spaces the assertion {\em is} true.
We answer this question
by showing that it holds precisely for a class of spaces closely
related to $H^2$.
\
Nicholas Young
28 October 1999.